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A fundamental facility location problem is to choose the location of facilities, such as industrial plants and warehouses, to minimize the cost of satisfying the demand for some commodity. There are associated costs for locating the facilities, as well as transportation costs for distributing the commodities. We assume that the transportation costs form a metric. This problem is commonly referred to as the uncapacitated facility location (UFL) problem. Applications to bank account location and clustering, as well as many related pieces of work, are discussed by Cornuejols, Nemhauser and Wolsey [2]. Recently, the first constant factor approximation algorithm for this problem was obtained by Shmoys, Tardos and Aardal [16].We show that a simple greedy heuristic combined with the algorithm by Shmoys, Tardos and Aardal, can be used to obtain an approximation guarantee of 2.408. We discuss a few variants of the problem, demonstrating better approximation factors for restricted versions of the problem.We also show that the problem is Max SNP-hard. However, the inapproximability constants derived from the Max SNP hardness are very close to one. By relating this problem to Set Cover, we prove a lower bound of 1.463 on the best possible approximation ratio assuming N P / ∈ DT IM E[n O(log log n) ].

Abstract-In this paper we present a clustering scheme to create a hierarchical control structure for multi-hop wireless networks. A cluster is defined as a subset of vertices, whose induced graph is connected. In addition, a cluster is required to obey certain constraints that are useful for management and scalability of the hierarchy. All these constraints cannot be met simultaneously for general graphs, but we show how such a clustering can be obtained for wireless network topologies. Finally, we present an efficient distributed implementation of our clustering algorithm for a set of wireless nodes to create the set of desired clusters.

We develop a new randomized rounding approach for fractional vectors defined on the edge-sets of bipartite graphs. We show various ways of combining this technique with other ideas, leading to improved (approximation) algorithms for various problems. These include: -low congestion multi-path routing; -richer random-graph models for graphs with a given degree-sequence; -improved approximation algorithms for: (i) throughput-maximization in broadcast scheduling, (ii) delay-minimization in broadcast scheduling, as well as (iii) capacitated vertex cover; and -fair scheduling of jobs on unrelated parallel machines.By the induction hypothesis, Pr [S 1 ∪ S 2 ∪ {x}] ≤ Pr [S 1 ∪ {x}]Pr [S 2 ] and hence (6) follows. An identical argument holds for the case where p 1 , q 1 ∈ S 2 .Case 3. p 1 ∈ S 1 and q 1 ∈ E \ (S 1 ∪ S 2 ). Let S 1 = S 1 \ {p 1 }. Exactly one of two events happens during the iteration.Event A. p 1 gets rounded to zero. In this case,By the induction hypothesis, Pr [S 1 ∪ S 2 ] ≤ Pr [S 1 ]Pr [S 2 ] and hence (6) follows. Event B. p 1 does not get rounded to zero. In this case, Pr[S 1 | T ] = Pr [S 1 ∪ {x}] Pr[S 2 | T ] = Pr [S 2 ] Pr[S 1 ∪ S 2 | T ] = Pr [S 1 ∪ S 2 ∪ {x}]. By the induction hypothesis, Pr [S 1 ∪ S 2 ∪ {x}] ≤ Pr [S 1 ∪ {x}]Pr [S 2 ] and hence (6) follows. Case 4. { p 1 , q 1 } ⊆ E \(S 1 ∪ S 2 ). In this case, Pr[S 1 | T ] = Pr [S 1 ], Pr[S 2 | T ] = Pr [S 2 ] and Pr[S 1 ∪ S 2 | T ] = Pr [S 1 ∪ S 2 ]; we are done by the induction hypothesis. This completes the proof of Theorem 2.4.